%-------------------------------------------------------------------------------

% This file is part of Code_Saturne, a general-purpose CFD tool.
%
% Copyright (C) 1998-2013 EDF S.A.
%
% This program is free software; you can redistribute it and/or modify it under
% the terms of the GNU General Public License as published by the Free Software
% Foundation; either version 2 of the License, or (at your option) any later
% version.
%
% This program is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
% FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
% details.
%
% You should have received a copy of the GNU General Public License along with
% this program; if not, write to the Free Software Foundation, Inc., 51 Franklin
% Street, Fifth Floor, Boston, MA 02110-1301, USA.

%-------------------------------------------------------------------------------

% Name of classical tools of the Code_Saturne environment
% (note that Code_Saturne itself is defined in csvers.tex)
%
%\usepackage{color}
%\usepackage[dvipsnames]{xcolor}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%EDF's Coulors
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% GREEN MF
%\definecolor{greenmf}{RGB}{0,128,0}
%% RED MF
%\definecolor{redmf}{RGB}{128,0,0}
%% Bleu edf
\definecolor{blueedf}{RGB}{0,91,187}
%% Orange edf
%\definecolor{orangeedf}{RGB}{255,160,47}
%% Bleu edf
%\definecolor{bluededf}{RGB}{9,53,122}
%% Orange edf
%\definecolor{orangededf}{RGB}{254,88,21}
%% Green edf
%\definecolor{greenedf}{RGB}{80,158,47}
%\definecolor{greendedf}{RGB}{196,214,47}
%% Manchester
%\definecolor{indigo}{RGB}{106,44,145}
%\definecolor{gold}{RGB}{255,215,0}
%\definecolor{mag}{RGB}{255,0,239}
%%

%\newcommand{\ensight}{EnSight\xspace}
%\newcommand{\ensightg}{EnSight Gold\xspace}
%\newcommand{\fluent}{FLUENT\xspace}
%\newcommand{\bft}{BFT\xspace}
%\newcommand{\catalyst}{Catalyst\xspace}
%\newcommand{\fvm}{FVM\xspace}
%\newcommand{\gambit}{GAMBIT\xspace}
%\newcommand{\gmsh}{Gmsh\xspace}
%\newcommand{\harpoon}{Harpoon\xspace}
%\newcommand{\hexpress}{\emph{HexPress}\xspace}
%\newcommand{\icemcfd}{ICEM~CFD\xspace}
%\newcommand{\ideas}{I-deas\xspace}
%\newcommand{\med}{MED\xspace}
%\newcommand{\metis}{M\textsc{etis}\xspace}
%\newcommand{\parmetis}{P\textsc{ar}M\textsc{etis}\xspace}
%\newcommand{\nopo}{{\sc nopo}\xspace}
%\newcommand{\paraview}{ParaView\xspace}
%\newcommand{\pcs}{Preprocessor\xspace}
%\newcommand{\salome}{SALOME\xspace}
%\newcommand{\scotch}{S\textsc{cotch}\xspace}
%\newcommand{\ptscotch}{PT-S\textsc{cotch}\xspace}
%\newcommand{\simail}{S\textsc{IMAIL}\xspace}
%\newcommand{\starcd}{\textbf{STAR-CD}\xspace}
%\newcommand{\starccmp}{\textbf{STAR-CCM+}\xspace}
%\newcommand{\syrthes}{SYRTHES\xspace}
%\newcommand{\vtk}{VTK\xspace}
%\newcommand{\CA}{%
%{\fontfamily{ppl}\fontshape{it}\selectfont Code\_Aster}\xspace}
%\newcommand{\CS}{
%{\fontfamily{ppl}\fontshape{it}\selectfont Code\_Saturne}\xspace}
%\newcommand{\CN}{
%{\fontfamily{ppl}\fontshape{it}\selectfont Neptune\_CFD}\xspace}
%\newcommand{\codits}{%
%{\fontfamily{ppl}\fontshape{it}\selectfont Codits.f90}\xspace}
%
% VnV macros
%
%\newcommand{\vnvitem}{\item[\hspace{1cm} -]}
%\newcommand{\vnvcase}[1][]{\item[#1]}
%%
% Mathematical operators (and other useful related macros)
%
\newcommand{\dive}{\text{div}\,}
\newcommand{\divs}{\text{div}\,}
\newcommand{\divv}{\underline{\text{div}}\,}
\newcommand{\divt}{\underline{\underline{\text{div}}}\,}
\newcommand{\grad}{\underline{\nabla}}
\newcommand{\gradback}[3]{\nabla^{\text{back}} \left(#1\right)_{#2}^{#3}}
\newcommand{\gradcent}[3]{\nabla^{\text{cent}} \left(#1\right)_{#2}^{#3}}
\newcommand{\gradcentsq}[3]{(\nabla^2)^{\text{cent}} \left(#1\right)_{#2}^{#3}}
\newcommand{\diffu}[3]{\Delta \left(#1\right)_{#2}^{#3}}
\newcommand{\diffusq}[3]{(\Delta^2) \left(#1\right)_{#2}^{#3}}
\newcommand{\graddiffu}[3]{(\nabla^{\text{cent}}\Delta) \left(#1\right)_{#2}^{#3}}
\newcommand{\hess}{\underline{\nabla}^2}
\newcommand{\ggrad}{\text{g}\underline{\underline{\text{rad}}}\,}
\newcommand{\gradv}{\underline{\nabla}}
\newcommand{\gradt}{\underline{\underline{\nabla}}}
\newcommand{\gradtt}{\underline{\underline{\underline{\nabla}}}}
\newcommand{\mat}[1]{\underline{\textrm{#1}}}
\newcommand{\matt}[1]{\underline{\underline{\textrm{#1}}}}
\newcommand{\rot}{\underline{\text{rot}}\ }
\newcommand{\vect}[1]{\underline{#1}}
%\newcommand{\tens}[1]{\underline{\underline{#1}}}
\newcommand{\transpose}[1]{#1^T}
\newcommand{\symmetric}[1]{#1^S}
\newcommand{\trace}{tr}
\newcommand{\deviator}[1]{#1^D}
\newcommand{\norm}[1]{\left| #1 \right|}
\newcommand{\nnorm}[1]{\left|\left| #1 \right|\right|}
%
% Discretization
%
\newcommand{\Facei}[1]{\mathcal{F}^{int}_{#1}} % set of internal faces
\newcommand{\Faceb}[1]{\mathcal{F}^{ext}_{#1}} %set of boundary faces
\newcommand{\Faceib}[1]{\mathcal{F}^{}_{#1}} %set of boundary faces
\newcommand{\Face}[1]{\mathcal{F}_{#1}} % set of faces
\newcommand{\face}{{f}} % face
\newcommand{\fij}{{f_{ij}}} % internal face
\newcommand{\fib}{{f_{b}}} % boundary face
\newcommand{\iface}{{f_\celli}} % orientated face
\renewcommand{\ij}{{ij}} % orientated internal face value ("K, sigma" in sushi)
\newcommand{\ji}{{ji}} % orientated internal face value ("L, sigma" in sushi)
\newcommand{\ib}{{ib}} % orientated internal face value ("K, sigma" in sushi)
\newcommand{\celli}{i} % name of the current cell
\newcommand{\centi}{I} % center of the current cell
\newcommand{\centip}{{I^\prime}} % orthogonal center of the current cell
\newcommand{\centipp}{{I^{\prime\prime}}} %
\newcommand{\ipf}{{I^\prime}} % orthogonal center indice of the current cell
\newcommand{\vol}[1]{\Omega_{#1}} % Volume of the cell #1 (K itself in sushi)
\newcommand{\spaceaverage}[2]{\frac{1}{\norm{\Omega_{#1}}}\int_{\Omega_{#1}}\,#2\,d\Omega} % Space average
\newcommand{\spav}[2]{\overline{#1}_{#2}} %space average abreviation
\newcommand{\discspav}[3]{\overline{#1}_{#2}^{#3}} %space average abreviation
\newcommand{\timeaverage}[2]{\frac{1}{t_{#2}-t_{#1}}\int_{t_{#1}}^{t_{#2}}} % Time average
\newcommand{\timeaveragecl}[2]{\frac{1}{\Delta t}\int_{t_{#1}}^{t_{#2}}} % Time average classic
\newcommand{\tiav}[2]{#1^{#2}} % time average abreviation
\newcommand{\densityaverage}[2]{\frac{\spav{#1}{#2}}{\spav{\rho}{#2}} } % Density average
\newcommand{\densav}[2]{\widetilde{#1}_{#2}} %Density average abreviation
\newcommand{\sumboundaries}[1]{\displaystyle \sum_{\face \in \Faceib{\celli}} \left( \int_{\face}#1~dS \right)\cdot{\vect n}_{\face}}

\newcommand{\sumboundariesdisc}[1]{\displaystyle\sum_{\face \in \Faceib{\celli}} #1\cdot{\vect S}_{\face}}

\newcommand{\massflux}[2]{\dot{m}_{#1}^{#2}}

\newcommand{\discfluxoned}[2]{{\color{blue}g_{#1}^{#2}}}

\newcommand{\fluxoned}[1]{{\color{blue}g \left(#1 \right)}}

\newcommand{\sumboundariesdiscmf}[1]{\displaystyle\sum_{\face \in \Faceib{\celli}} #1 \massflux{\face}{n,n+1}}


\newcommand{\sumboundariesdiscmfint}[1]{\displaystyle\sum_{\face \in \Facei{\celli}} #1 \massflux{\face}{n,n+1}}
\newcommand{\sumboundariesdiscmfintplus}[1]{\displaystyle\sum_{\face \in \Facei{\celli}} \left[ #1 \massflux{\face}{n,n+1}\right]^+}

\newcommand{\sumboundariesdiscmfintminus}[1]{\displaystyle\sum_{\face \in \Facei{\celli}} \left[ #1 \massflux{\face}{n,n+1}\right]^-}

\newcommand{\sumboundariesdiscmfsign}[2]{\displaystyle\sum_{\face \in \Faceib{\celli}/~\massflux{\face}{n,n+1}#2} #1 \massflux{\face}{n,n+1}}

\newcommand{\sumboundariesdiscgrdpasign}[2]{\displaystyle\sum_{\face \in \Faceib{\celli}/~\massflux{\face}{n,n+1}#2} #1 \underline{S}_{\face}}

\newcommand{\sumboundariesinternaldiscmfsign}[2]{\displaystyle\sum_{j,#2} #1 \massflux{\face}{n,n+1}}

\newcommand{\sumboundariesborderdiscmfsign}[3]{\displaystyle\sum_{b^{#2},\massflux{\face}{n,n+1}#3} #1 \massflux{\face}{n,n+1}}

\newcommand{\cellj}{j} % name of the neighbouring face
\newcommand{\centj}{J} % center of the neighbouring face
\newcommand{\centjp}{{J^\prime}} % orthogonal center of the neighbouring cell
\newcommand{\centjpp}{{J^{\prime\prime}}} %
\newcommand{\jpf}{{J^\prime}} % orthogonal center indice of the neighbouring cell
\newcommand{\centf}{F} % center of the face
\newcommand{\cento}{O} % intersection between the cell centers and the face

%%%%%%%%%% LOW MACH MACRO %%%%%%%%%%%%%

%% SIGLES %%%

\newcommand{\Epx}{\textit{Europlexus}}

%%%%%%%%%%%%%%%%

\newcommand{\epssq}{\mathscr{E}_0^2}
\newcommand{\eps}{\mathscr{E}_0}
\newcommand{\epsinf}{\mathscr{E}_{inf}}
\newcommand{\bareps}{\overline{\mathscr{E}}_0}
\newcommand{\Mmax}{M_{max}}
%\newcommand{\MSmax}{M_{max}^S}
\newcommand{\MSmax}{M_{S,\,max}}
\newcommand{\Mmin}{M_{min}}
\newcommand{\Mthres}{M_{thres}}
\newcommand{\Minf}{M_{inf}}
\newcommand{\gammaeps}{\gamma_{\eps}}
\newcommand{\epsm}{\varepsilon^{-}}
\newcommand{\rhoeps}{\rho^{\varepsilon}}

%%%% DIPHASIQUE %%%%%%
\newcommand{\pl}{p_{\text{\normalfont liq}}}
\newcommand{\pv}{p_{\text{\normalfont vap}}}
\newcommand{\vell}{u_{\text{\normalfont liq}}}
\newcommand{\velv}{u_{\text{\normalfont vap}}}
\newcommand{\psat}{p^{\text{\normalfont sat}}}
\newcommand{\Tl}{T_{\text{\normalfont liq}}}
\newcommand{\Tv}{T_{\text{\normalfont vap}}}
\newcommand{\Vl}{v_{\text{\normalfont liq}}}
\newcommand{\Machl}{M_{\text{\normalfont liq}}}
\newcommand{\Machv}{M_{\text{\normalfont vap}}}
\newcommand{\Vvap}{v_{\text{\normalfont vap}}}
\newcommand{\Tsat}{T^{\text{\normalfont sat}}}
\newcommand{\taul}{\tau_{\text{\normalfont liq}}}
\newcommand{\tauv}{\tau_{\text{\normalfont vap}}}
\newcommand{\hl}{h_{\text{\normalfont liq}}}
\newcommand{\hv}{h_{\text{\normalfont vap}}}
\newcommand{\rhol}{\rho_{\text{\normalfont liq}}}
\newcommand{\rhov}{\rho_{\text{\normalfont vap}}}
\newcommand{\mul}{\mu_{\text{\normalfont liq}}}
\newcommand{\muv}{\mu_{\text{\normalfont vap}}}
\newcommand{\el}{\varepsilon_\text{\normalfont liq}}
\newcommand{\ev}{\varepsilon_\text{\normalfont vap}}
\newcommand{\elsat}{\varepsilon_\text{\normalfont liq}^{\text{\normalfont sat}}}
\newcommand{\evsat}{\varepsilon_\text{\normalfont vap}^{\text{\normalfont sat}}}
\newcommand{\taulsat}{\tau_\text{\normalfont liq}^{\text{\normalfont sat}}}
\newcommand{\tauvsat}{\tau_\text{\normalfont vap}^{\text{\normalfont sat}}}
\newcommand{\rholsat}{\rho_\text{\normalfont liq}^{\text{\normalfont sat}}}
\newcommand{\rhovsat}{\rho_\text{\normalfont vap}^{\text{\normalfont sat}}}
\newcommand{\gamlsat}{\gamma_\text{\normalfont liq}^{\text{\normalfont sat}}}
\newcommand{\gamvsat}{\gamma_\text{\normalfont vap}^{\text{\normalfont sat}}}
\newcommand{\al}{\alpha_{\text{\normalfont liq}}}
\newcommand{\av}{\alpha_{\text{\normalfont vap}}}
\newcommand{\alsat}{\alpha_{\text{\normalfont liq}}^{\text{\normalfont sat}}}
\newcommand{\avsat}{\alpha_{\text{\normalfont vap}}^{\text{\normalfont sat}}}
\newcommand{\kapvsat}{\kappa_{\text{\normalfont vap}}^{\text{\normalfont sat}}}
\newcommand{\kaplsat}{\kappa_{\text{\normalfont liq}}^{\text{\normalfont sat}}}
\newcommand{\sslsat}{c_\text{\normalfont liq}^{\text{\normalfont sat}}}
\newcommand{\ssvsat}{c_\text{\normalfont vap}^{\text{\normalfont sat}}}

\newcommand{\Elya}{\mathcal{E}}

\newcommand{\Cour}{\mathscr{C}}
\newcommand{\CourAeps}{\mathscr{C}_{\Aeps}}
\newcommand{\CourCeps}{\mathscr{C}_{\Ceps}}
\newcommand{\CourC}{\mathscr{C}_{\norm{u}}}
\newcommand{\CourA}{\mathscr{C}_{\norm{u}+c}}
\newcommand{\CourCth}{\mathscr{C}_{\norm{u}}^{th}}
\newcommand{\CourCcrit}{\mathscr{C}_{\norm{u}}^{\text{\normalfont crit}}}
\newcommand{\CourE}{\mathscr{C}_{E}}
\newcommand{\CourS}{\mathscr{C}_{S}}

\newcommand{\Mshock}[1]{M_{#1}^{\text{\normalfont S}}}
\newcommand{\sigshock}[1]{\sigma_{#1}^{\text{\normalfont S}}}

\newcommand{\CourCu}{\mathscr{C}_{u_0}}
\newcommand{\CourCue}{\mathscr{C}_{\frac{u_0}{\epsilon}}}

\newcommand{\SRC}{\norm{G}_{\Ceps}}
\newcommand{\SRA}{\norm{G}_{\Aeps}}
\newcommand{\SRAC}{\norm{G}_{\Aeps\,\Ceps}}
\newcommand{\GC}{\mathbf{G}_{\Ceps}}
\newcommand{\GA}{\mathbf{G}_{\Aeps}}

\newcommand{\Ceps}{\mathcal{C}}
\newcommand{\Aeps}{\mathcal{A}}
\newcommand{\ALP}{\mathcal{A}_{LP}}
\newcommand{\sC}{s_{\Ceps}}

\newcommand{\Am}{\mathbf{A}_{0}}
\newcommand{\Acm}{\mathbf{A}^{\mathcal{C}}_{0}}
\newcommand{\Aam}{\mathbf{A}^{\mathcal{A}}_{0}}
\newcommand{\Bm}{\mathbf{B}_{0}}
\newcommand{\Bcm}{\mathbf{B}^{\mathcal{C}}_{0}}
\newcommand{\Bam}{\mathbf{B}^{\mathcal{A}}_{0}}
\newcommand{\Rm}{\mathbf{R}_{0}}
\newcommand{\Rcm}{\mathbf{R}^{\mathcal{C}}_{0}}
\newcommand{\Ram}{\mathbf{R}^{\mathcal{A}}_{0}}
\newcommand{\Lambm}{\mathbf{\Lambda}_{0}}
\newcommand{\Lambcm}{\mathbf{\Lambda}^{\mathcal{C}}_{0}}
\newcommand{\Lambam}{\mathbf{\Lambda}^{\mathcal{A}}_{0}}
\newcommand{\Pcm}{\mathbf{P}^{\mathcal{C}}_{0}}
\newcommand{\Pam}{\mathbf{P}^{\mathcal{A}}_{0}}
\newcommand{\difc}{\lambda_0^{\mathcal{C}}}
\newcommand{\difa}{\lambda_0^{\mathcal{A}}}
\newcommand{\dif}{\lambda_0}
\newcommand{\Uv}{\mathbf{U}}
\newcommand{\Uvt}{\mathbf{\overset{\mathbf{\sim}}{\Uv}}}
\newcommand{\Vv}{\mathbf{V}}
\newcommand{\Vvm}{\mathbf{V_m}}
\newcommand{\Hv}{\mathbf{H}}
\newcommand{\Hspe}[1]{\Hv_{#1}}
\newcommand{\dHspe}[2]{\mathbf{\delta H}_{#1}^{#2}}
\newcommand{\Hvt}{\widetilde{\mathbf{H}}}
\newcommand{\Fv}{\mathbf{F}}
\newcommand{\Fvcal}{\mathbf{\mathcal{F}}}
\newcommand{\fv}{\mathbf{f}}
\newcommand{\Fcv}{\mathbf{F}^{\mathcal{C}}}
\newcommand{\Fceps}{\mathbf{F}^{\Ceps}}
\newcommand{\Faeps}{\mathbf{F}^{\Aeps}}
\newcommand{\Dpaeps}{\mathcal{D}_p^{\Aeps}}
\newcommand{\Duaeps}{\mathcal{D}_u^{\Aeps}}
\newcommand{\Fav}{\mathbf{F}^{\mathcal{A}}}
\newcommand{\Wv}{\mathbf{W}}
\newcommand{\Sv}{\mathbf{S}}
\newcommand{\SvRel}{\textcolor{orange}{\mathbf{S}}}
\newcommand{\wv}{\mathbf{w}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\vvFT}{\mathbf{\hat{v}}}
\newcommand{\Zv}{\mathbf{Z}}

\newcommand{\omc}{\omega_{0}^{\mathcal{C}}}
\newcommand{\oma}{\omega_{0}^{\mathcal{A}}}

\newcommand{\omcs}{\omega_{\mathcal{C}}^0}
\newcommand{\omas}{\omega_{\mathcal{A}}^0}

\newcommand{\Pxi}{\mathbf{\mathcal{P}}(\xi)}
\newcommand{\Prxi}{\mathbf{\mathcal{P}^{R}}(\xi)}
\newcommand{\Dm}{\mathbf{D}_0}
\newcommand{\Drm}{\mathbf{D}^{R}_0}

\newcommand{\Herm}[1]{\mathcal{H}\left(#1\right)}

\newcommand{\gradphim}{\grad\phi\left(\Uv_0\right)}
\newcommand{\gradphiminv}{\left(\grad\phi\left(\Uv_0\right)\right)^{-1}}

\newcommand{\vel}{\mathbf{u}}
\newcommand{\Id}{\mathbf{I}}

\newcommand{\Frelax}[1]{\vect{\mathscr{F}}_{#1}}
\newcommand{\Fc}[1]{\vect{F}^{\mathcal{C}_{#1}}}
\newcommand{\Fcrelax}[1]{\vect{\mathscr{F}}^{\mathcal{C}_{#1}}}
\newcommand{\Fa}[1]{\vect{F}^{\mathcal{A}_{#1}}}
\newcommand{\Farelax}[1]{\vect{\mathscr{F}}^{\mathcal{A}_{#1}}}
\newcommand{\Un}[1]{\vect{U}_{#1}^n}
\newcommand{\Fsch}[2]{\Fv^{#2}_{#1}}

%scheme
\newcommand{\sch}[3]{\mathbf{#1}_{#2}^{#3}}
\newcommand{\scht}[3]{\mathbf{\widetilde{#1}}_{#2}^{#3}}
\newcommand{\im}{\celli-1}
\newcommand{\ip}{\celli+1}
\newcommand{\ifm}{\celli-1/2}
\newcommand{\ifp}{\celli+1/2}

%Hyperbolic system

\newcommand{\eigv}[2]{\lambda_{#1}^{#2}}
\newcommand{\eigveq}[2]{\lambda_{#1}^{\text{\normalfont eq} #2}}
\newcommand{\eigr}[2]{\mathbf{r}_{#1}^{#2}}
\newcommand{\eigl}[2]{\mathbf{l}_{#1}^{#2}}
\newcommand{\eigvEuler}[1]{\lambda_{#1}^{\text{\normalfont Euler}}}
\newcommand{\cv}{\Ceps}
\newcommand{\ac}{\Aeps}
\newcommand{\lambdaco}{\lambda_{\text{\normalfont u}}}
\newcommand{\lambdaac}{\lambda_{\text{\normalfont u+c}}}

%Relaxation Scheme

\newcommand{\aI}{a_{\Ceps}}
\newcommand{\aII}{a_{\Aeps}}
\newcommand{\aLP}{a_{LP}}
\newcommand{\aE}{a_{E}}
\newcommand{\RI}[3]{\mathcal{I}_{#1,\,#2}^{#3}}
\newcommand{\RIp}[1]{\mathcal{I}_{#1}}
\newcommand{\RIpt}[1]{\widetilde{\mathcal{I}}_{#1}}
\newcommand{\s}[2]{s_{#1}^{#2}}
\newcommand{\dx}{\Delta x}
\newcommand{\xV}{\mathbf{x}}
\newcommand{\dt}{\Delta t}
\newcommand{\dts}{\delta t}
\newcommand{\dtbar}{\overline{\Delta t}}
\newcommand{\dtC}{\Delta t_{\Ceps}}
\newcommand{\dtA}{\Delta t_{\Aeps}}
\newcommand{\dtLP}{\Delta t_{\text{\normalfont LP}}}
\newcommand{\dtALP}{\Delta t_{\ALP}}
\newcommand{\dtE}{\Delta t_{E}}
\newcommand{\dtT}{\Delta t_{\mathcal{T}}}
\newcommand{\dm}{\Delta m}

\newcommand{\PiRel}{\textcolor{orange}{\Pi}}
\newcommand{\muRel}{\textcolor{orange}{\mu}}
\newcommand{\aIRel}{\textcolor{orange}{\aI}}
\newcommand{\aIIRel}{\textcolor{orange}{\aII}}
\newcommand{\WRel}{\textcolor{orange}{W}}
\newcommand{\RRel}{\textcolor{orange}{R}}
%
\newcommand{\WSAzero}{WSA\textsuperscript{0}\,}
\newcommand{\pI}{(I)\,}
\newcommand{\pII}{(II)\,}
\newcommand{\pIb}{\textcolor{blue}{(I)}\,}
\newcommand{\pIIb}{\textcolor{blue}{(II)}\,}
\newcommand{\pointi}{\textcolor{blue}{(i)}\,}
\newcommand{\pointii}{\textcolor{blue}{(ii)}\,}
\newcommand{\pointiii}{\textcolor{blue}{(iii)}\,}
\newcommand{\obji}{\textcolor{blue}{1.}\,}
\newcommand{\objii}{\textcolor{blue}{2.}\,}

%continuous
\newcommand{\cont}[3]{\vect{#1}_{#2}^{#3}}
\newcommand{\contt}[3]{\vect{\widetilde{#1}}_{#2}^{#3}}

\newcommand{\ustar}[1]{u^*_{#1}}
\newcommand{\pstar}[1]{\Pi^*_{#1}}
\newcommand{\Pstar}[1]{p^*_{#1}}
\newcommand{\pstartheta}[1]{\Pi^{*,\,\theta}_{#1}}
\newcommand{\taustarC}[1]{\tau^*_{#1,\,\Ceps}}
\newcommand{\taustarA}[1]{\tau^*_{#1,\,\Aeps}}
\newcommand{\rhostarC}[1]{\rho^*_{#1,\,\Ceps}}
\newcommand{\rhostarA}[1]{\rho^*_{#1,\,\Aeps}}
\newcommand{\estarC}[1]{e^*_{#1,\,\Ceps}}
\newcommand{\epsstarC}[1]{\varepsilon^*_{#1,\,\Ceps}}
\newcommand{\estarA}[1]{e^*_{#1,\,\Aeps}}
\newcommand{\epsstarA}[1]{\varepsilon^*_{#1,\,\Aeps}}

\newcommand{\SL}[1]{S_L^{#1}}
\newcommand{\SR}[1]{S_R^{#1}}

\newcommand{\rhoSL}[1]{\rho_L\,S_L^{#1}}
\newcommand{\rhoSR}[1]{\rho_R\,S_R^{#1}}

\newcommand{\ZstarC}[1]{\Zv^*_{#1,\,\Ceps}}
\newcommand{\ZstarstarC}[1]{\Zv^{**}_{#1,\,\Ceps}}
\newcommand{\UstarC}[1]{\Uv^*_{#1,\,\Ceps}}
\newcommand{\UstarstarC}[1]{\Uv^{**}_{#1,\,\Ceps}}
\newcommand{\ZstarA}[1]{\Zv^*_{#1,\,\Aeps}}
\newcommand{\ZstarstarA}[1]{\Zv^{**}_{#1,\,\Aeps}}
\newcommand{\UstarA}[1]{\Uv^*_{#1,\,\Aeps}}
\newcommand{\UstarstarA}[1]{\Uv^{**}_{#1,\,\Aeps}}
\newcommand{\ZstarALP}[1]{\Zv^*_{#1,\,\ALP}}
\newcommand{\ZstarstarALP}[1]{\Zv^{**}_{#1,\,\ALP}}

\newcommand{\Ustar}[1]{\sch{U}{#1}{*,\,n}}
\newcommand{\Ustarstar}[1]{\sch{U}{#1}{**,\,n}}
\newcommand{\Wstar}[1]{\sch{W}{#1}{*,\,n}}
\newcommand{\Wstarstar}[1]{\sch{W}{#1}{**,\,n}}

%numerical weights for relaxation

\newcommand{\wgtp}[1]{\omega_{\ifp}^{#1,\,n}}
\newcommand{\wgtm}[1]{\omega_{\ifm}^{#1,\,n}}

\newcommand{\epsp}{\epsilon}
\newcommand{\epspx}{\epsilon\,\dx}

%Conservative variable

\newcommand{\Meq}{\mathscr{M}}
\newcommand{\Req}{\mathscr{R}}
\newcommand{\rhoU}{\rho\,u}
\newcommand{\rhoUsq}{\rho\,u^2}
\newcommand{\internalEOS}{\varepsilon^{eos}(\rho,\, p)}
\newcommand{\rhointernal}{\rho\,\varepsilon}
\newcommand{\rhointernalEOS}{\rho\,\varepsilon^{eos}(\rho,\, p)}
\newcommand{\pU}{p\,u}

\newcommand{\cEOS}{c^{eos}\left(\rho ,\, p \right)}
\newcommand{\pEOS}{p^{eos}\left(\rho ,\, \varepsilon \right)}
\newcommand{\cEOSI}{c_{\Ceps}}
\newcommand{\cEOSIz}{(c_{\Ceps})_0}
\newcommand{\cEOSII}{c_{\Aeps}}
\newcommand{\cEOSIIz}{(c_{\Aeps})_0}

\newcommand{\SoundSpeedI}{\sqrt{\left(\der{\varepsilon}{p}_{|\rho}\right)^{-1}\,\left(\epssq\frac{p}{\rho^2}-\der{\varepsilon}{\rho}_{|p} \right)}}

\newcommand{\SoundSpeedII}{\sqrt{\left(\der{\varepsilon}{p}_{|\rho}\right)^{-1} \epssq\frac{p}{\rho^2}}}

\newcommand{\RhoSoundSpeedSq}[1]{\left(\der{\varepsilon}{p}_{|\rho}\right)^{-1}\,\left(#1\,\frac{p}{\rho}-\rho\,\der{\varepsilon}{\rho}_{|p} \right)}

\newcommand{\RhoSoundSpeedIISq}{\left(\der{\varepsilon}{p}_{|\rho}\right)^{-1}\,\frac{p}{\rho} }

\newcommand{\PS}[1]{\Phi_{#1}}

%Adimentionalization

\newcommand{\rhobar}{\overline{\rho}}
\newcommand{\ubar}{\overline{u}}
\newcommand{\pbar}{\overline{p}}

\newcommand{\rhoZref}{\rho_{\text{\normalfont ref}}^0}
\newcommand{\uZref}{u_{\text{\normalfont ref}}^0}
\newcommand{\pZref}{p_{\text{\normalfont ref}}^0}
\newcommand{\MZref}{M_{\text{\normalfont ref}}^0}

\newcommand{\Lref}{l_{\text{\normalfont ref}}}
\newcommand{\Tref}{t_{\text{\normalfont ref}}}
\newcommand{\Tconv}{\tau_{\text{\normalfont conv}}}
\newcommand{\Rhoref}{\rho_{\text{\normalfont ref}}}
\newcommand{\Betaref}{\beta_{\text{\normalfont ref}}}
\newcommand{\Uref}{u_{\text{\normalfont ref}}}
\newcommand{\Pref}{p_{\text{\normalfont ref}}}
\newcommand{\Celref}{c_{\text{\normalfont ref}}}
\newcommand{\phiref}{\phi_{\text{\normalfont r}}}
\newcommand{\St}{S\text{\normalfont \normalfont t}_\text{\normalfont \normalfont r}}

\newcommand{\Mref}[1]{m_{\text{\normalfont r,\,#1}}}
\newcommand{\MUref}[1]{{(m u)}_{\text{\normalfont r,\,#1}}}

\newcommand{\Psat}{P^{\text{\normalfont sat}}}
\newcommand{\rhosat}[1]{\rho_{#1}^{\text{\normalfont sat}}}
\newcommand{\tausat}[1]{\tau_{#1}^{\text{\normalfont sat}}}
\newcommand{\Ysat}[1]{Y_{#1}^{\text{\normalfont sat}}}
\newcommand{\ssat}[1]{s_{#1}^{\text{\normalfont sat}}}
\newcommand{\asat}[1]{\alpha_{#1}^{\text{\normalfont sat}}}
\newcommand{\cvspe}[1]{(c_{\text{\normalfont v}})_{#1}}

%Thermodynamics

\newcommand{\Pinf}{P_{\infty}}

%Discrete Flux Notations

%Space

\newcommand{\Flux}[2]{\mathbb{F}_{#1}^{#2}}
\newcommand{\U}[2]{\mathbb{U}_{#1}^{#2}}
\newcommand{\facem}{L}
\newcommand{\facep}{R}
\newcommand{\facepm}{R\,L}
\newcommand{\ik}{i_{k}}
\newcommand{\duLR}{u_R-u_L}
\newcommand{\dpLR}{p_R-p_L}
\newcommand{\Hrelax}{\mathscr{H}}

\newcommand{\xp}{x^{\prime}}
%\newcommand{\up}{u^{\prime}}
\newcommand{\upri}{u^{\prime}}
\newcommand{\ut}{\tilde{u}}
\newcommand{\uha}{\hat{u}}
\newcommand{\cha}{\hat{c}}

%\newcommand{\Wb}{\mathbf{W}}
%\newcommand{\Rb}{\mathbf{R}}
\newcommand{\Wb}{W}
\newcommand{\Rb}{R}
\newcommand{\Wt}[1]{W_{#1}(t^n,\,t^{\nd})}
\newcommand{\Rt}[1]{R_{#1}(t^n,\,t^{\nd})}
\newcommand{\taub}{\mathbf{\tau}}
\newcommand{\thetap}{\theta_0^{p}}
\newcommand{\thetau}{\theta_0^{u}}

\newcommand{\rhopert}[2]{\rho_{#1}^{#2}}
\newcommand{\upert}[2]{u_{#1}^{#2}}
\newcommand{\ppert}[2]{p_{#1}^{#2}}
\newcommand{\pipert}[2]{\Pi_{#1}^{#2}}
\newcommand{\epert}[2]{e_{#1}^{#2}}

\newcommand{\AW}{\mathbf{\vect{\vect{A}}}^\Wb}
\newcommand{\AR}{\mathbf{\vect{\vect{A}}}^\Rb}
\newcommand{\Ustateb}{\mathbf{\vect{U}}}

%Time
\newcommand{\nd}{n+}
\newcommand{\nC}{\widetilde{n}}
\newcommand{\nup}{n+1/2}
\newcommand{\Tend}{T_{\text{\normalfont end}}}

%Numerical Results
\newcommand{\order}[2]{\mathcal{O}\left( #1\right)_{#2}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%  BAER NUNZIATO  %%%%%%%%%%%%%

\newcommand{\mk}[1]{m_{#1}}
\newcommand{\I}{\text{\normalfont I}}
\newcommand{\uIs}{u_\text{\normalfont I}^*}
\newcommand{\DeltajumpAuIs}{\left[\Avec_k \right]_{\uIs}}
\newcommand{\Cdist}{\mathbf{C}_0^{\infty}\left(\mathbb{R}\times [0,\,+\infty[ \right)^p}
\newcommand{\Cone}{\mathbf{C}^{1}\left(\mathbb{R}\times [0,\,+\infty[ \right)^p}
\newcommand{\DiruIs}{\delta_{\left\{x= \uIs\,t \right\}}}
\newcommand{\phib}{\bm{\phi}}
\newcommand{\Drift}{\mathscr{D}}
\newcommand{\Hsour}{\mathscr{H}}
\newcommand{\Qsour}{\mathscr{Q}}
\newcommand{\Usour}{\mathscr{U}}
\newcommand{\uDsour}{\widetilde{u}}

%rho_k*c_k^2
\newcommand{\Cs}[1]{C_{#1}}
\newcommand{\Csp}[1]{C^{'}_{#1}}

\newcommand{\duV}{\Delta u}
\newcommand{\dpV}{\Delta p}

\newcommand{\duVp}{\Delta u^{\prime}}
\newcommand{\dpVp}{\Delta p^{\prime}}

\newcommand{\cw}{c_{\text{\normalfont w}}}
\newcommand{\cstar}{c_{\text{\normalfont star}}}
\newcommand{\cfr}{c_{\text{\normalfont froz}}}

%%%% Relaxation Times  %%%%%%

\newcommand{\tauP}{\tau_{\text{\normalfont p}}}
\newcommand{\tauU}{\tau_{\text{\normalfont u}}}
\newcommand{\tauM}{\tau_{\text{\normalfont m}}}
\newcommand{\tauT}{\tau_{\text{\normalfont t}}}

%\newcommand{\rhopert}{\rho_{\text{\normalfont pert}}}
%\newcommand{\ppert}{p_{\text{\normalfont pert}}}

\newcommand{\Kp}{K_{\text{\normalfont p}}}
\newcommand{\Kpt}{\widetilde{K}_{\text{\normalfont p}}}
\newcommand{\Kptp}{\left(\widetilde{K}_{\text{\normalfont p}}\right)^{'}}
\newcommand{\Ku}{K_{\text{\normalfont u}}}
\newcommand{\Kut}{\widetilde{K}_{\text{\normalfont u}}}
\newcommand{\Kutp}{\left(\widetilde{K}_{\text{\normalfont u}}\right)^{'}}

\newcommand{\etaP}{\eta_{\text{\normalfont p}}}
\newcommand{\etaU}{\eta_{\text{\normalfont u}}}

\newcommand{\adu}{\alpha_{\duV}}
\newcommand{\adp}{\alpha_{\dpV}}
\newcommand{\ad}[1]{\alpha_{#1}}
\newcommand{\htil}{\widetilde{h}}

\newcommand{\Wa}{W^{\alpha}}
\newcommand{\WP}{W^{P}}
\newcommand{\Wdp}{W^{\dpV}}

%%%%%  TWO PHASE EOS  %%%%%%%%%%%

\newcommand{\phiEOS}[2]{#1_{#2}^{\text{\normalfont EOS}}}
\newcommand{\PhaseSpaceBN}{\Phi^\Cour}
\newcommand{\PhaseSpaceBNInfty}{\Phi_{\Pinf}^\Cour}
\newcommand{\InvariantBNInfty}{\mathcal{I}_{\Pinf}^\Cour}
\newcommand{\PhaseSpaceBNInftyClosed}{\overline{\Phi}_{\Pinf}^\Cour}
\newcommand{\PhaseSpaceBNElya}{\overline{\Phi}_{\Elya_\infty}}
\newcommand{\diff}{\mathrm{d}}

%%%%%%% MATRICES for non conservative systems

\newcommand{\pprime}{\prime\prime}
\newcommand{\ttil}{\widetilde{t}}
\newcommand{\AreaL}{\mathcal{A}_L}
\newcommand{\AreaR}{\mathcal{A}_R}
\newcommand{\AreaStar}{\mathcal{A}^*}

\newcommand{\Zeromat}{\underline{\underline{\mathbf{{0}}}}}
\newcommand{\Zerovec}{\underline{\mathbf{{0}}}}
\newcommand{\Dmat}{\underline{\underline{\mathbf{{D}}}}}
\newcommand{\Jmat}{\underline{\underline{\mathbf{{J}}}}}
\newcommand{\Amat}{\underline{\underline{\mathbf{{A}}}}}
\newcommand{\deltamat}{\underline{\underline{\mathbf{{\delta}}}}}
\newcommand{\Rmat}{\underline{\underline{\mathbf{{R}}}}}
\newcommand{\Emat}{\underline{\underline{\mathbf{{E}}}}}
\newcommand{\Mmat}{\underline{\underline{\mathbf{{M}}}}}
\newcommand{\Pmat}{\underline{\underline{\mathbf{{P}}}}}
\newcommand{\Qmat}{\underline{\underline{\mathbf{{Q}}}}}
\newcommand{\Ematt}{\underline{\underline{\mathbf{{\widetilde{E}}}}}}
\newcommand{\Ematst}{\underline{\underline{\mathbf{{E}}}}^*}
\newcommand{\Smat}{\underline{\underline{\mathbf{{S}}}}}
\newcommand{\Svec}{\mathbf{{S}}}
\newcommand{\Hvec}{\mathbf{{H}}}
\newcommand{\Avec}{\mathbf{{a}}}
\newcommand{\Kvec}{\mathbf{{k}}}
\newcommand{\Bvec}{\mathbf{{b}}}
\newcommand{\Cvec}{\mathbf{{c}}}
\newcommand{\Dvec}{\mathbf{{d}}}
\newcommand{\Mv}{\mathbf{{M}}}
\newcommand{\Pv}{\mathbf{{P}}}
\newcommand{\Pvec}{\mathbf{p}}
\newcommand{\Mvk}[1]{\mathbf{{M}}_{\alpha,\,#1}}
\newcommand{\Pvk}[1]{\mathbf{{P}}_{\alpha,\,#1}}
\newcommand{\Xv}{\mathbf{{X}}}
\newcommand{\Yvec}{\mathbf{{y}}}
\newcommand{\Xvec}{\mathbf{{x}}}
\newcommand{\Zvec}{\mathbf{{z}}}
\newcommand{\Avt}{\overset{\mathbf{\sim}}{\Avec}}

\newcommand{\Deltav}{\mathbf{{\Delta}}}
\newcommand{\Deltavec}{\underline{\Delta}}
\newcommand{\Ucons}{\mathbf{\mathcal{U}}}
\newcommand{\Phitau}{\Phi_\tau}
\newcommand{\Phitaumat}{\underline{\underline{\Phi}}_\tau}
\newcommand{\Psitau}{\Psi_\tau}
\newcommand{\Psitaumat}{\underline{\underline{\Psi}}_\tau}

\newcommand{\dirac}[1]{\mathbf{\delta}_{\left\{#1\right\}}}
\newcommand{\gammah}{\hat{\gamma}}

\newcommand{\Cmat}{\underline{\underline{\mathbf{{C}}}}}
\newcommand{\Taumat}{\underline{\underline{\mathbf{{\tau}}}}}
\newcommand{\Gvec}{\mathbf{{G}}}
\newcommand{\lvec}[1]{\mathbf{{l}}_{#1}}
\newcommand{\Gvect}{\widetilde{\mathbf{{G}}}}

\newcommand{\Kmat}[1]{\underline{\underline{\mathbf{{K}}}}_{#1}}
\newcommand{\Krelax}{\underline{\underline{\mathbf{{K}}}}^{\text{\normalfont relax}}}
\newcommand{\Kconv}{\underline{\underline{\mathbf{{K}}}}^{\text{\normalfont conv}}}
\newcommand{\KAmat}{\Kmat{\mathbf{A}}}

\newcommand{\Omat}[1]{\underline{\underline{\mathbf{{O}}}}_{#1}}

\newcommand{\Bmat}{\underline{\underline{\mathbf{{B}}}}}
\newcommand{\Bmatr}{\underline{\underline{\mathbf{B}}}^{\mathbf{r}}}
\newcommand{\Bmatd}{\underline{\underline{\mathbf{B}}}^{\mathbf{d}}}
\newcommand{\Bmatdelta}{\delta \underline{\underline{\mathbf{B}}}}
\newcommand{\Bmateq}{\underline{\underline{\mathbf{{B}}}}^{\text{\normalfont eq}}}
\newcommand{\Rmateq}{\underline{\underline{\mathbf{{R}}}}^{\text{\normalfont eq}}}
\newcommand{\Lambdamateq}{\underline{\underline{\mathbf{\Lambda}}}^{\text{\normalfont eq}}}
\newcommand{\Lambdamat}{\underline{\underline{\mathbf{\Lambda}}}}
\newcommand{\Lambdavec}{\underline{\mathbf{\Lambda}}}
\newcommand{\Bmatrelax}{\underline{\underline{\mathbf{{B}}}}^{\text{\normalfont relax}}}

\newcommand{\diag}[1]{\underline{\underline{\text{\normalfont diag}}}\left(#1 \right)}
\newcommand{\tp}{t^{'}}

\newcommand{\Cmateq}{\underline{\underline{\mathbf{{C}}}}^{\text{\normalfont eq}}}
\newcommand{\CmateqAsym}{\underline{\underline{\mathbf{{C}}}}^{\text{\normalfont eq}}_{\Delta \rightarrow 0}}

\newcommand{\Gveceq}{\mathbf{{G^{\text{\normalfont eq}}}}}
\newcommand{\Seq}{\mathscr{S}^{\text{\normalfont eq}}}
\newcommand{\Ueq}{\Uv^{\text{\normalfont eq}}}
\newcommand{\Weq}{\Wv^{\text{\normalfont eq}}}
\newcommand{\Uvref}{\Uv^{\text{\normalfont ref}}}
\newcommand{\Wvref}{\Wv^{\text{\normalfont ref}}}
\newcommand{\Aveceq}{\Avec^{\text{\normalfont eq}}}
\newcommand{\Preseq}{\mathcal{P}^{\text{\normalfont eq}}}


\newcommand{\wap}[2]{\omega_{#1}^{#2}}
\newcommand{\gap}[2]{\omega_{#1,\,#2}}
\newcommand{\Bap}[2]{B_{#1,\,#2}}
\newcommand{\gapt}[2]{\hat{\omega}_{#1,\,#2}}

\newcommand{\mclinsq}{\alpha_1 \Cs{1}+ \alpha_2 \Cs{2}}




%%%%%%%%%% REPORT MACRO %%%%%%%%%%%%%%%


\newcommand{\cunr}{C^{1}\left(\mathbb{R} \right)}
\newcommand{\cdeuxr}{C^{2}\left(\mathbb{R} \right)}
\newcommand{\cunrr}{C^{1}\left(\mathbb{R},\mathbb{R}^{+} \right)}
\newcommand{\cdeuxrr}{C^{2}\left(\mathbb{R},\mathbb{R}^{+} \right)}

\newcommand{\Top}[1]{T_{k,\Delta x}^{\Delta t}\left( #1\right)}
\newcommand{\Hop}[1]{H_{k,\Delta x}^{\Delta t}\left( #1\right)}
\newcommand{\NHop}[3]{H_{#2,#3}^{\Delta t}\left( #1\right)}
\newcommand{\TFHop}[3]{\widehat{H}_{#2,#3}^{\Delta t}\left( #1\right)}

\newcommand{\entrop}{S}
\newcommand{\F}{F}
\newcommand{\entroptwo}[1]{S\left( #1\right)}
\newcommand{\G}[1]{G\left( #1\right)}

\newcommand{\trinorm}[1]{||| #1 |||}

\newcommand{\schemecol}[1]{{\color{blue} #1}}

\newcommand{\M}[2]{\mathbb{M}_{#1}^{#2}}
%\newcommand{\E}[2]{\mathbb{E}_{#1}^{#2}}
\newcommand{\C}[2]{\mathbf{C}_{#1}^{#2}}
%\newcommand{\D}[2]{\mathbf{D}_{#1}^{#2}}

\newcommand{\transpvel}[2]{c_{#1}^{#2}}

\newcommand{\cstdeltam}{\delta^{-}}
\newcommand{\cstdeltaM}{\delta^{+}}
\newcommand{\variastar}{\varia^{*}}
\newcommand{\variatheta}[1]{\varia^{#1+\thetascheme}}

\newcommand{\philim}[1]{{\color{blue}\phi\left( #1\right)}}
\newcommand{\sourceterm}[1]{\mathbf{\Gamma}_{#1}}
\newcommand{\visc}[1]{\mathbf{K}_{#1}}

\newcommand{\IMP}[1]{\mathbf{IMP}\left(#1 \right)}
\newcommand{\EXP}[1]{\mathbf{EXP}\left(#1 \right)}

\newcommand{\cons}[1]{CONS\left(#1 \right)}

\newcommand{\variahe}{Y_{He}}

\newcommand{\Ncells}{N_{cells}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\dderiv}[2]{\dfrac{\mathrm{d}#2}{\mathrm{d}#1}} % total derivative
\newcommand{\dd}{\mathrm{d}} % total derivative
%\newcommand{\der}[2]{\dfrac{\partial #1}{\partial #2}}
\newcommand{\der}[2]{\partial_{#2}\,#1}
\newcommand{\dder}[2]{\dfrac{\partial^2 #1}{\partial #2^2}}
\newcommand{\dderd}[3]{\dfrac{\partial^2 #1}{\partial #2 \partial #3}}
%
\newcommand{\varia}{Y} % variable name
\newcommand{\variatilde}{\widetilde{Y}} % variable name
\newcommand{\variasup}{Y_{sup}} % variable name
\newcommand{\variainf}{Y_{inf}} % variable name
\newcommand{\discvaria}[2]{ {\color{red}Y_{#1}^{#2}}} %
\newcommand{\discvariacheck}[2]{ {\color{red}\check{Y}_{#1}^{#2}}} %
\newcommand{\Deltavaria}[3]{\Delta^{#3}\discvaria{#1}{#2}
}
\newcommand{\discvariatilde}[2]{ {\color{red}\widetilde{Y}_{#1}^{#2}}} %

\newcommand{\discvariatwo}[2]{ {\color{red}V_{#1}^{#2}}} %

\newcommand{\discvariavec}[2]{ {\mathbf{Y}_{#1}^{#2}}} % variable name
\newcommand{\vdiscvaria}[1]{ {\color{red}\vect{Y}^{#1}}} % variable name
\newcommand{\discvariavectilde}[2]{ {\color{red}\vect{\widetilde{Y}}_{#1}^{#2}}} % variable name
\newcommand{\TFdiscvaria}[2]{ {\color{red}\widehat{Y}_{#1}^{#2}}} % variable name
\newcommand{\deltadiscvaria}[2]{ {\color{red}\delta Y_{#1}^{#2}}} % variable name
\newcommand{\vdeltadiscvaria}[1]{ {\color{red}\vect{\delta Y}^{#1}}} % variable name
\newcommand{\deltadiscvariatilde}[2]{ {\color{red}\delta \widetilde{Y}_{#1}^{#2}}} % variable name
\newcommand{\thetascheme}{{\color{orange}\theta}}
\newcommand{\betascheme}[1]{{\color{blue}\beta_{#1}}}
\newcommand{\betaschemeinf}[1]{{\color{blue}\beta_{#1}^{inf}}}
\newcommand{\betaschemesup}[1]{{\color{blue}\beta_{#1}^{max}}}
\newcommand{\betaschememin}[1]{{\color{blue}\beta_{#1}^{min}}}
\newcommand{\betaschememax}[1]{{\color{blue}\beta_{#1}^{max}}}
\newcommand{\betaschememinst}[1]{{\color{blue}\beta_{#1}^{min,*}}}
\newcommand{\betaschememaxst}[1]{{\color{blue}\beta_{#1}^{max,*}}}
\newcommand{\discvariaF}[2]{ {\color{blue} Y_{#1}^{#2}} } % variable name
\newcommand{\discvariaFtilde}[2]{ {\color{blue} \widetilde{Y}_{#1}^{#2}} } % variable name
\newcommand{\discvariaFtheta}[2]{ {\color{blue}#2 Y_{#1}^{n+1}+(1-#2)Y_{#1}^{n}} } % variable name
\newcommand{\courant}[2]{\nu_{#1}^{#2}}

\newcommand{\fsimp}{\underbrace{\left(\alpha_i^n-\sum\limits_{\face \in \Faceib{\celli}}\massflux{\face}{n} \right)}_{\alpha_{\celli}^{n+1}}}

\newcommand{\fsimpdiag}{\left(\underbrace{\alpha_i^n-\sum\limits_{\face \in \Faceib{\celli}}\massflux{\face}{n}}_{\alpha_{\celli}^{n+1}}+\thetascheme \sum\limits_{\face \in \Faceib{\celli}~/\massflux{\face}>0}\massflux{\face_{\ij}}{n}  \right)}

\newcommand{\fsexp}{\discvaria{\celli}{n}\left(  \sum\limits_{\face \in \Faceib{\celli}}\massflux{\face}{n}\right)}

\newcommand{\EpsilonOp}[1]{ \fsimp \left(#1_{\celli}-\discvaria{\celli}{n}\right)+\thetascheme \left( \sum\limits_{\face \in \Faceib{\celli}}{\color{blue}#1_{\face}}\massflux{\face}{n} \right)+(1-\thetascheme)\left( \sum\limits_{\face \in \Faceib{\celli}}{\discvariaF{\face}{n}}\massflux{\centf}{n}\right)-\fsexp}

\newcommand{\EpsilonOpsecond}[1]{ \alpha_i^n \left(#1_{\celli}-\discvaria{\celli}{n}\right)+\thetascheme \left( \sum\limits_{\face \in \Faceib{\celli}}({\color{blue}#1_{\face}}-#1_{\celli})\massflux{\face}{n} \right)+(1-\thetascheme)\left( \sum\limits_{\face \in \Faceib{\celli}}{(\discvariaF{\face}{n}-#1_{\celli})\massflux{\face}{n}}\right)}

\newcommand{\EmOp}[1]{ \fsimp #1_{\celli}+\thetascheme \left( \sum\imits_{\face \in \Faceib{\celli}}{\color{blue}#1_{\face}^{U}}\massflux{\face}{n} \right)}

\newcommand{\EmOpsecond}[1]{ \alpha_{\celli}^{n} #1_{\celli}+ \sum\limits_{\face \in \Faceib{\celli}}\left(\thetascheme{\color{blue}#1_{\face}^{U}}-#1_{\celli}\right)\massflux{\face}{n} }

\newcommand{\variav}{\vect{v}} % vectorial variable name
%\newcommand{\variat}{\tens{t}} % vectorial variable name
\newcommand{\ncell}{{N_{cell}}} % number of cells
%
\newcommand{\Neigh}[1]{Neigh\left(#1\right)}
\newcommand{\Bound}[1]{Bound\left(#1\right)}
%
\newcommand{\doxygenfile}[2]{\href{run:./doxygen/src/#1}{#2}}
%
\newcommand{\bm}[1]{\text{\normalfont \boldmath $#1$ \unboldmath} \! \!}
\newcommand{\comp}[1]{\textrm{#1}}
\newcommand{\degresC}{$^\circ$C}
% TODO remove
\newcommand{\etape}[1]{\vspace{0.3cm}$\bullet\ ${\bf #1}\\}
%
\newcommand{\fort}[1]{\texttt{#1}}
\newcommand{\ind}[1]{\text{\normalfont $_{#1}$}}
\newcommand{\Max}{\text{\normalfont Max}}
\newcommand{\Min}{\text{\normalfont Min}}
\newcommand{\nl}{\vspace{1ex}}
% TODO remove
\newcommand{\var}[1]{\ensuremath{\text{\normalfont \texttt{#1}}}}
%

% Definition of variables and keywords (used in the different tables)
%
\newbox\tempbox
\newcommand{\motcle}[7]{%
   \noindent
   \setbox\tempbox\hbox{\hspace*{2.5cm}}
   \makebox[2.5cm][l]{\tt #1}\index{#1}\makebox[1.3cm][l]{#2}\makebox[6.cm][l]{#3}%
   \makebox[4.cm][l]{[#4]}#5\hspace{1cm}#6\\
   \hangindent\wd\tempbox\quad\ignorespaces#7\bigskip}
\newcommand{\motcleb}[7]{%
   \noindent
   \setbox\tempbox\hbox{\hspace*{2.5cm}}
   \makebox[2.5cm][l]{\tt \bf #1}\index{#1}\makebox[1.3cm][l]{#2}\makebox[6.cm][l]{#3}%
   \makebox[4.cm][l]{[#4]}#5\hspace{1cm}#6\\
   \hangindent\wd\tempbox\quad\ignorespaces#7\bigskip}
%
\newcommand{\variab}[4]{%
       \setbox\tempbox\hbox{ifacel [e]: }
       \hangindent\wd\tempbox\noindent{{\tt #2\index{#1}} [#3]:\quad}\ignorespaces#4.}
\newcommand{\variabsize}[2]{%
   \noindent
   \makebox[.12\linewidth][l]{\tt #1:}\index{#1}\parbox[t]{.88\linewidth}{#2.}}
\newcommand{\variablist}[4]{%
       \setbox\tempbox\hbox{ifacel [e]: }
       \hangindent\wd\tempbox\noindent{{\tt #2\index{#1}} [#3]:\quad}\ignorespaces#4\quad}
\newcommand{\envvar}[1]{\medskip\noindent{\ttfamily \large #1}}
%
% Other useful macros (for \raggedright in tabular, for a Unix tilde)
%
\newcommand{\PreserveBackslash}[1]{\let\temp=\\#1\let\\=\temp}
\newcommand{\tildeunix}{{\huge$_{_{\widetilde{\ }}}$}\hspace*{0.1mm}}
